Limit cycles appearing from the perturbation of a system with a multiple line of critical points
type: Item
instance of: scholarly article
title: Limit cycles appearing from the perturbation of a system with a multiple line of critical points
zbMATH Open document ID: 1241.34037
author: Armengol Gasull
author: Chengzhi Li
author: Joan Torregrosa
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-11-03T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/150536
review text: Consider the family of planar systems \[ \dot{x}=-yC(x,y)+\epsilon P(x,y);~\,\dot{y}=xC(x,y)+\epsilon Q(x,y) \] where \(P,Q\) and \(C\) are real polynomials, \(C(0,0) \neq 0\), and \(\epsilon\) is a small real parameter. The number of zeros of the abelian integral \(M(r)\) on \(x^{2}+y^{2}=r^{2}\) controls the number of limit cycles that bifurcate from the periodic orbits of the unperturbed system with \(\epsilon =0\). The authors consider the case \(C(x,y)=(1-y)^{m}\) and \(P, Q\) are of degree \(n\). They prove that \(M(r)=[\frac{m+n}{2}]-1\) when \(n<m-1\) and \(n\) when \(n \geq m-1\).
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5967364
zbMATH Keywords: limit cycles
zbMATH Keywords: weak Hilbert 16th problem
zbMATH Keywords: abelian integrals
zbMATH Keywords: bifurcation of periodic orbits
MaRDI profile type: Publication
OpenAlex ID: W2029563761
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DOI: 10.1016/J.NA.2011.08.032
description: scientific article; zbMATH DE number 5967364